Exercise 8

Added Exercise 8 to repository

Informatik_I/Exercise_8/Task_1/README.org

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+#+title: Task 1: Reverse Digits
+
+#+author: JirR02
+* Task
+:PROPERTIES:
+:CUSTOM_ID: task
+:END:
+The goal of this task is to implement a program to read a number and
+print its reverse.
+
+You should complete the implementation of the reverse function in
+=reverse.cpp=. This function gets an =int= as input parameter, and it
+should print the digits of the number in reverse.
+
+The =reverse= function must be implemented recursively (without any
+loop). To enforce this, we disallow the use of =for= or =while=.
+
+* Input
+:PROPERTIES:
+:CUSTOM_ID: input
+:END:
+The input is a single non-negative =int= number.
+
+An input example:
+
+#+begin_src shell
+  321231
+#+end_src
+
+* Output
+:PROPERTIES:
+:CUSTOM_ID: output
+:END:
+For the above input, the output should be:
+
+#+begin_src shell
+  132123
+#+end_src
+
+Note: You are not allowed to use the string library to solve this task!
+
+* Solution
+:PROPERTIES:
+:CUSTOM_ID: solution
+:END:
+#+begin_src cpp
+#include "reverse.h"
+#include <iostream>
+
+// PRE: n >= 0
+// POST: print the digits of the input in reverse order to std::cout
+void reverse(int n) {
+
+  int res = n % 10;
+  std::cout << res;
+
+  int new_int = n / 10;
+
+  if (new_int == 0) {
+    return;
+  }
+
+  reverse(new_int);
+}
+#+end_src

Informatik_I/Exercise_8/Task_2/README.org

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+#+title: Task 2: Set Product
+
+#+author: JirR02
+* Task
+:PROPERTIES:
+:CUSTOM_ID: task
+:END:
+In this exercise, you are going to implement a function which computes
+the /n-fold Cartesian product/ for sets. The n-fold Cartesian product is
+a way to combine multiple sets of items. Imagine you have multiple sets,
+denoted as \(A_1, A_2, ..., A_n\). The n-fold Cartesian product is like
+taking one item from each set and combining them into a new set.
+
+For example, let's say you have two sets:
+
+\[
+A_1 = \{a,b,c\} \text{ and } A_2 = \{ x, y, z \}
+\]
+
+The Cartesian product of these two sets would be a new set that contains
+all possible combinations of one item from \(A_1\) and one item from
+\(A_2\):
+
+\[
+A_1 \times A_2 = \{(a,x),(a,y),(a,z),(b,x),(b,y),(b,z),(c,x),(c,y),(c,z)\}
+\]
+
+In general, for any two sets \(A\) and \(B\), the Cartesian product
+\(A times B\) is defined as:
+
+\[
+A \times B \{(a,b) | a \in A \text{ and } b \in B\}
+\]
+
+This can be generalized to the n-fold Cartesian product. For any \(n\)
+sets \(A_1, A_2, ... , A_n\), the n-fold Cartesian product is defined
+as:
+
+\[
+A_1 \times A_2 \times ... \times A_n = \{(a_1, a_2, ..., a_n) | a_i \in A_i \text{ for all } i = 1, 2, ... , n\}
+\]
+
+Note: This task naturally lends itself to a *recursive implementation*.
+
+** The Set class
+:PROPERTIES:
+:CUSTOM_ID: the-set-class
+:END:
+A set class is implemented in =set.h= and =set.ipp=. This class
+implements a number of useful operations on sets. *For an overview of
+its functionalities, please refer to this week's Power Set code
+example*. Note that you do not need to understand any of the code in
+=set.h= and =set.ipp= in order to use it!
+
+* Input & Output
+:PROPERTIES:
+:CUSTOM_ID: input-output
+:END:
+When the program starts, you are first prompted to enter the number of
+sets. Then, you are prompted to enter each set. Sets are entered on a
+single line using the format
+=<number of char elements> <elements separated by spaces>=.
+
+Once all sets are entered, the set product is computed using the
+=set_product= function. Finally, the resulting set is printed to the
+console.
+
+*** Example
+:PROPERTIES:
+:CUSTOM_ID: example
+:END:
+#+begin_src shell
+Number of sets: 2
+Set 0: 3 a b c
+Set 1: 2 x y
+Product Set:
+{ax,ay,bx,by,cx,cy}
+#+end_src
+
+*Note*: Each input set is a set of char elements. At least 1 set must be
+inputted.
+
+* Solution
+:PROPERTIES:
+:CUSTOM_ID: solution
+:END:
+#+begin_src cpp
+#include <iterator>
+#include <string>
+#include <vector>
+
+#include "set.h"
+#include "solution.h"
+
+StringSet set_product(const std::vector<CharSet> &sets) {
+  StringSet res;
+  StringSet ram;
+  if (sets.size() == 1) {
+    for (int i = 0; i < sets[0].size(); ++i) {
+      std::string j(1, sets[0][i]);
+    res.insert(j);
+    }
+  return res;
+  }
+
+  CharSet first_subset = sets[0];
+  std::vector<CharSet> remaining_set(sets.begin() + 1, sets.end());
+
+  StringSet res_subset = set_product(remaining_set);
+
+  for (char char_first_subset : first_subset.elements()) {
+    for (std::string str_res_subset : res_subset.elements()) {
+    res.insert(std::string(1, char_first_subset) + str_res_subset);
+    }
+  }
+
+  return res;
+}
+#+end_src

Informatik_I/Exercise_8/Task_3/README.org

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+#+title: Task 3: All permutations
+
+#+author: JirR02
+* Task
+:PROPERTIES:
+:CUSTOM_ID: task
+:END:
+In this exercise, you are going to implement a function which computes
+the set of all permutations of a given string.
+
+Note: This task naturally lends itself to a *recursive implementation*.
+
+** Example
+:PROPERTIES:
+:CUSTOM_ID: example
+:END:
+The string "abc" has the following set of permutations:
+
+{ abc, acb, bac, bca, cab, cba }
+
+** The Set class
+:PROPERTIES:
+:CUSTOM_ID: the-set-class
+:END:
+A set class is implemented in =set.h= and =set.ipp=. This class
+implements a number of useful operations on sets. *For an overview of
+its functionalities, please refer to the Power Set code example*. Note
+that you do not need to understand any of the code in =set.h= and
+=set.ipp= in order to use it!
+
+* Input & Output
+:PROPERTIES:
+:CUSTOM_ID: input-output
+:END:
+When the program starts, you are first prompted to enter a string.
+
+Then, the program computes the set of all permutations using the
+function *all_permutations*. Finally, the resulting set is printed to
+the console.
+
+*** Example
+:PROPERTIES:
+:CUSTOM_ID: example-1
+:END:
+#+begin_src shell
+String: abc
+Permutations: 
+{abc,acb,bac,bca,cab,cba}
+#+end_src
+
+Hint: You may find the substr method useful. The method is documented
+here.
+
+* Solution
+:PROPERTIES:
+:CUSTOM_ID: solution
+:END:
+#+begin_src cpp
+#include "pair_sum.h"
+#include <climits>
+
+// PRE:  for any two indices i and j, v[i] + v[j] ≤ INT_MAX
+// POST: returns the number of indices (i,j), i<j of a vector v
+//       that corresponds to the given iterator range such that
+//       v[i] + v[j] == sum.
+int pairs_with_sum(int sum, iterator begin, iterator end) {
+  int count = 0;
+  int size = end - begin;
+
+  for (int i = 0; i < size; ++i) {
+    for (int j = i + 1; j < size; ++j) {
+      iterator cur_begin = begin + i;
+      iterator cur_end = begin + j;
+      if (*cur_begin + *cur_end == sum)
+  ++count;
+    }
+  }
+
+  return count;
+}
+#+end_src

Informatik_I/Exercise_8/Task_4/README.org

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+#+title: Task 4: Dual Numbers
+
+#+author: JirR02
+* Dual Numbers
+:PROPERTIES:
+:CUSTOM_ID: dual-numbers
+:END:
+In this task, you are required to implement a basic version of /dual
+number/. Dual numbers are conceptually similar to complex number.
+
+A dual numbers \(n\) is of the following form:
+
+\[
+z = a + \epsilon b
+\]
+
+with the following identity:
+
+\[
+\epsilon ^2 = 0
+\]
+
+Adding two dual numbers is done component-wise:
+
+\[
+(a + \epsilon b) + (c + \epsilon d) + (a+c) + \epsilon (b + d)
+\]
+
+Subtracting two dual numbers is also done component-wise:
+
+\[
+(a + \epsilon b) - (c + \epsilon d) = (a + \epsilon b) + (-c - \epsilon d) = (a - c) + \epsilon(b - d)
+\]
+
+Using the identity above, multiplying two dual numbers gives
+
+\[
+(a + \epsilon b)(c + \epsilon d) = ac + \epsilon(ad + bc)
+\]
+
+** Tasks
+:PROPERTIES:
+:CUSTOM_ID: tasks
+:END:
+You are required to extend the provided implementation of dual numbers
+by adding the implementation for various arithmetic operators in
+=DualNumber.cpp=.
+
+*** Task 1: Addition operator
+:PROPERTIES:
+:CUSTOM_ID: task-1-addition-operator
+:END:
+Implement the addition operator
+
+#+begin_src shell
+DualNumber operator+(const DualNumber& dn1, const DualNumber& dn2)
+#+end_src
+
+in file =DualNumber.cpp=.
+
+*** Task 2: Subtraction operator
+:PROPERTIES:
+:CUSTOM_ID: task-2-subtraction-operator
+:END:
+Implement the subtraction operator
+
+#+begin_src shell
+DualNumber operator-(const DualNumber& dn1, const DualNumber& dn2)
+#+end_src
+
+in file =DualNumber.cpp=.
+
+*** Task 3: Multiplication operator
+:PROPERTIES:
+:CUSTOM_ID: task-3-multiplication-operator
+:END:
+Implement the multiplication operator
+
+#+begin_src shell
+DualNumber operator*(const DualNumber& dn1, const DualNumber& dn2)
+#+end_src
+
+in file =DualNumber.cpp=.
+
+* Solution
+:PROPERTIES:
+:CUSTOM_ID: solution
+:END:
+#+begin_src cpp
+#include "DualNumber.h"
+
+DualNumber operator+(const DualNumber &dn1, const DualNumber &dn2) {
+  DualNumber res;
+  double a = dn1.a + dn2.a;
+  double b = dn1.b + dn2.b;
+  res.a = a;
+  res.b = b;
+  return res;
+}
+
+DualNumber operator-(const DualNumber &dn1, const DualNumber &dn2) {
+  DualNumber res;
+  double a = dn1.a - dn2.a;
+  double b = dn1.b - dn2.b;
+  res.a = a;
+  res.b = b;
+  return res;
+}
+
+DualNumber operator*(const DualNumber &dn1, const DualNumber &dn2) {
+  DualNumber res;
+  double a = dn1.a * dn2.a;
+  double b = (dn1.a * dn2.b) + (dn1.b * dn2.a);
+  res.a = a;
+  res.b = b;
+  return res;
+}
+#+end_src